Quasi-isometry
The term quasi-isometry is used in mathematics to investigate the “rough” global geometry of metric spaces . It plays an important role in numerous areas of geometry, analysis and geometric group theory , for example in the theory of hyperbolic groups or in proving rigidity theorems.
Definitions
Be and two metric spaces . A (not necessarily continuous) mapping is a quasi-isometric embedding if there are constants and , so that
- .
The mapping is called quasi-tight, if a constant exists, so that it for each a are with
A quasi-isometry is a quasi-dense, quasi-isometric embedding.
Two mappings are finite apart if .
The spaces and are called quasi-isometric if there is a quasi-isometric drawing .
Examples
Every constrained metric space is quasi-isometric to the point.
The embedding is a quasi-isometry for the Euclidean metric on and . You can set , and in the above definition .
The finite at different generating systems , a group associated Cayley graph are quasi-isometric.
Švarc - Milnor -Lemma: If a finitely generated group acts co- compactly and actually discontinuously through isometrics on a Riemannian manifold , then (the Cayley graph of) is quasi-isometric to . (See also the Švarc-Milnor theorem .)
With one obtains from this in particular: The fundamental group of a compact Riemannian manifold is quasi-isometric to the universal superposition .
properties
- The identical mapping on a metric space is a quasi-isometry.
- The concatenation of quasi-isometric embeddings (quasi-isometrics) is again a quasi-isometric embedding (quasi-isometric).
- An image that has a finite distance from a quasi-isometric embedding (quasi-isometry) is again a quasi-isometric embedding (quasi-isometry).
- Two metric spaces and are quasi-isometric if and only if there are quasi-isometric embeddings and , so that both and and and are finite apart.
Categories
The metric spaces with the quasi-isometric embeddings form a category according to the above properties . However, this is not of interest for quasi-isometrics, since its isomorphisms must be bijective and therefore many important quasi-isometries are not isomorphisms, such as the quasi-isometry between and mentioned in the examples above .
One therefore moves to a category in which the metric spaces are still the objects, but the morphisms are equivalence classes of quasi-isometric embeddings. Two quasi-isometric embeddings are called equivalent if they have a finite distance, and this obviously defines an equivalence relation. If the equivalence class denotes the quasi-isometric embedding , the definitions result
- ( Well-defined !)
a category. In this category, the isomorphisms are exactly the equivalence classes of quasi-isometries. The automorphism group of a metric space formed in this category is called its quasi-isometric group.
literature
- Clara Löh: Geometric group theory, an introduction. Script for the lecture Geometric Group Theory at the University of Regensburg, 2015. (English; PDF; 1.3 MB), Chapter 5.
- Michael Kapovich : Lectures in quasi-isometric rigidity. (Engl .; PDF; 319 kB).
Individual evidence
- ^ Clara Löh: Geometric Group Theory, Springer-Verlag 2017, ISBN 978-3-319-72253-5 , definition 5.1.6
- ^ Clara Löh: Geometric Group Theory, Springer-Verlag 2017, ISBN 978-3-319-72253-5 , Proposition 5.1.10
- ↑ Clara Löh: Geometric Group Theory, Springer-Verlag 2017, ISBN 978-3-319-72253-5 , 5.1.12 and 5.1.13